Questions

Page: 28
Section: 1.A
Number: 1

Suppose

and

are real numbers, not both

. Find real numbers

and

such that

Page: 28
Section: 1.A
Number: 2

Show that

is a cube root of

(meaning that its cube equals

).

Page: 28
Section: 1.A
Number: 3

Find two distinct square roots of

.

Page: 28
Section: 1.A
Number: 4

Show that

for all

.

Page: 28
Section: 1.A
Number: 5

Show that

for all

.

Page: 28
Section: 1.A
Number: 6

Show that

for all

.

Page: 28
Section: 1.A
Number: 7

Show that for every

, there exists a unique

such that

.

Page: 28
Section: 1.A
Number: 8

Show that for every

with

, there exists a unique

such that

.

Page: 28
Section: 1.A
Number: 9

Show that

for all

.

Page: 28
Section: 1.A
Number: 10

Find

such that

Page: 28
Section: 1.A
Number: 11

Explain why there does not exist

such that

Page: 28
Section: 1.A
Number: 12

Show that

for all

.

Page: 28
Section: 1.A
Number: 13

Show that

for all

and all

.

Page: 28
Section: 1.A
Number: 14

Show that

for all

.

Page: 28
Section: 1.A
Number: 15

Show that

for all

and all

.

Page: 28
Section: 1.A
Number: 16

Show that

for all

and all

.

Page: 34
Section: 1.B
Number: 1

Prove that

for every

.

Page: 34
Section: 1.B
Number: 2

Suppose

, and

. Prove that

or

.

Page: 34
Section: 1.B
Number: 3

Suppose

. Explain why there exists a unique

such that

.

Page: 34
Section: 1.B
Number: 4

The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in 1.19. Which one?

Page: 34
Section: 1.B
Number: 5

Show that in the definition of a vector space (1.19), the additive inverse condition can be replaced with the condition that

Here the

on the left side is the number

, and the

on the right side is the additive identity of

. (The phrase “a condition can be replaced” in a definition means that the collection of objects satisfying the definition is unchanged if the original condition is replaced with the new condition.)

Page: 34
Section: 1.B
Number: 6

Let

and

denote two distinct objects, neither of which is in

. Define an addition and scalar multiplication on

as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for

define

Is

a vector space over

? Explain.

Page: 41
Section: 1.C
Number: 1

For each of the following subsets of

, determine whether it is a subspace of

: (a)

; (b)

; ©

; (d)

.

Page: 41
Section: 1.C
Number: 2

Verify all the assertions in Example 1.35.

Page: 41
Section: 1.C
Number: 3

Show that the set of differentiable real-valued functions

on the interval

such that

is a subspace of

.

Page: 41
Section: 1.C
Number: 4

Suppose

. Show that the set of continuous real-valued functions

on the interval

such that

is a subspace of

if and only if

.

Page: 41
Section: 1.C
Number: 5

Is

a subspace of the complex vector space

?

Page: 41
Section: 1.C
Number: 6

(a) Is

a subspace of

? (b) Is

a subspace of

?

Page: 41
Section: 1.C
Number: 7

Give an example of a nonempty subset

of

such that

is closed under addition and under taking additive inverses (meaning

whenever

), but

is not a subspace of

.

Page: 41
Section: 1.C
Number: 8

Give an example of a nonempty subset

of

such that

is closed under scalar multiplication, but

is not a subspace of

. SECTION 1.C Subspaces

Page: 41
Section: 1.C
Number: 9

A function

is called periodic if there exists a positive number

such that

for all

. Is the set of periodic functions from

to

a subspace of

? Explain.

Page: 41
Section: 1.C
Number: 10

Suppose

and

are subspaces of

. Prove that the intersection

is a subspace of

.

Page: 41
Section: 1.C
Number: 11

Prove that the intersection of every collection of subspaces of

is a subspace of

.

Page: 41
Section: 1.C
Number: 12

Prove that the union of two subspaces of

is a subspace of

if and only if one of the subspaces is contained in the other.

Page: 41
Section: 1.C
Number: 13

Prove that the union of three subspaces of

is a subspace of

if and only if one of the subspaces contains the other two. [This exercise is surprisingly harder than the previous exercise, possibly because this exercise is not true if we replace

with a field containing only two elements.]

Page: 41
Section: 1.C
Number: 14

Verify the assertion in Example 1.38.

Page: 41
Section: 1.C
Number: 15

Suppose

is a subspace of

. What is

?

Page: 41
Section: 1.C
Number: 16

Is the operation of addition on the subspaces of

commutative? In other words, if

and

are subspaces of

, is

?

Page: 41
Section: 1.C
Number: 17

Is the operation of addition on the subspaces of

associative? In other words, if

are subspaces of

, is

Page: 41
Section: 1.C
Number: 18

Does the operation of addition on the subspaces of

have an additive identity? Which subspaces have additive inverses?

Page: 41
Section: 1.C
Number: 19

Prove or give a counterexample: if

are subspaces of

such that

then

.

Page: 41
Section: 1.C
Number: 20

Suppose

Find a subspace

of

such that

.

Page: 41
Section: 1.C
Number: 26

CHAPTER

Vector Spaces

Page: 41
Section: 1.C
Number: 21

Suppose

Find a subspace

of

such that

.

Page: 41
Section: 1.C
Number: 22

Suppose

Find three subspaces

of

, none of which equals

, such that

.

Page: 41
Section: 1.C
Number: 23

Prove or give a counterexample: if

are subspaces of

such that

then

.

Page: 41
Section: 1.C
Number: 24

A function

is called even if

for all

. A function

is called

if

for all

. Let

denote the set of real-valued even functions on

and let

denote the set of real-valued odd functions on

. Show that

.

Page: 54
Section: 2.A
Number: 1

Suppose

spans

. Prove that the list

also spans

.

Page: 54
Section: 2.A
Number: 2

Verify the assertions in Example 2.18.

Page: 54
Section: 2.A
Number: 3

Find a number

such that

is not linearly independent in

.

Page: 54
Section: 2.A
Number: 4

Verify the assertion in the second bullet point in Example 2.20.

Page: 54
Section: 2.A
Number: 5

(a) Show that if we think of

as a vector space over

, then the list

is linearly independent. (b) Show that if we think of

as a vector space over

, then the list

is linearly dependent.

Page: 54
Section: 2.A
Number: 6

Suppose

is linearly independent in

. Prove that the list

is also linearly independent.

Page: 54
Section: 2.A
Number: 7

Prove or give a counterexample: If

is a linearly independent list of vectors in

, then

is linearly independent.

Page: 54
Section: 2.A
Number: 8

Prove or give a counterexample: If

is a linearly independent list of vectors in

and

with

, then

is linearly independent.

Page: 54
Section: 2.A
Number: 9

Prove or give a counterexample: If

and

are linearly independent lists of vectors in

, then

is linearly independent.

Page: 54
Section: 2.A
Number: 10

Suppose

is linearly independent in

and

. Prove that if

is linearly dependent, then

.

Page: 54
Section: 2.A
Number: 38

CHAPTER

Finite-Dimensional Vector Spaces

Page: 54
Section: 2.A
Number: 11

Suppose

is linearly independent in

and

. Show that

is linearly independent if and only if

Page: 54
Section: 2.A
Number: 12

Explain why there does not exist a list of six polynomials that is linearly independent in

.

Page: 54
Section: 2.A
Number: 13

Explain why no list of four polynomials spans

.

Page: 54
Section: 2.A
Number: 14

Prove that

is infinite-dimensional if and only if there is a sequence

of vectors in

such that

is linearly independent for every positive integer

.

Page: 54
Section: 2.A
Number: 15

Prove that

is infinite-dimensional.

Page: 54
Section: 2.A
Number: 16

Prove that the real vector space of all continuous real-valued functions on the interval

is infinite-dimensional.

Page: 54
Section: 2.A
Number: 17

Suppose

are polynomials in

such that

for each

. Prove that

is not linearly independent in

.

Page: 60
Section: 2.B
Number: 1

Find all vector spaces that have exactly one basis.

Page: 60
Section: 2.B
Number: 2

Verify all the assertions in Example 2.28.

Page: 60
Section: 2.B
Number: 3

(a) Let

be the subspace of

defined by

Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 60
Section: 2.B
Number: 4

(a) Let

be the subspace of

defined by

Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 60
Section: 2.B
Number: 5

Prove or disprove: there exists a basis

of

such that none of the polynomials

has degree

.

Page: 60
Section: 2.B
Number: 6

Suppose

is a basis of

. Prove that

is also a basis of

.

Page: 60
Section: 2.B
Number: 7

Prove or give a counterexample: If

is a basis of

and

is a subspace of

such that

and

, then

is a basis of

.

Page: 60
Section: 2.B
Number: 8

Suppose

and

are subspaces of

such that

. Suppose also that

is a basis of

and

is a basis of

. Prove that

is a basis of

.

Page: 65
Section: 2.C
Number: 1

Suppose

is finite-dimensional and

is a subspace of

such that

. Prove that

.

Page: 65
Section: 2.C
Number: 2

Show that the subspaces of

are precisely

, and all lines in

through the origin.

Page: 65
Section: 2.C
Number: 3

Show that the subspaces of

are precisely

, all lines in

through the origin, and all planes in

through the origin.

Page: 65
Section: 2.C
Number: 4

(a) Let

. Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 65
Section: 2.C
Number: 5

(a) Let

. Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 65
Section: 2.C
Number: 6

(a) Let

. Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 65
Section: 2.C
Number: 7

(a) Let

. Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 65
Section: 2.C
Number: 8

(a) Let

. Find a basis of

. (b) Extend the basis in part (a) to a basis of

. © Find a subspace

of

such that

.

Page: 65
Section: 2.C
Number: 9

Suppose

is linearly independent in

and

. Prove that

.

Page: 65
Section: 2.C
Number: 10

Suppose

are such that each

has degree

. Prove that

is a basis of

.

Page: 65
Section: 2.C
Number: 11

Suppose that

and

are subspaces of

such that

,

, and

. Prove that

. SECTION 2.C Dimension

Page: 65
Section: 2.C
Number: 12

Suppose

and

are both five-dimensional subspaces of

. Prove that

.

Page: 65
Section: 2.C
Number: 13

Suppose

and

are both 4-dimensional subspaces of

. Prove that there exist two vectors in

such that neither of these vectors is a scalar multiple of the other.

Page: 65
Section: 2.C
Number: 14

Suppose

are finite-dimensional subspaces of

. Prove that

is finite-dimensional and

Page: 65
Section: 2.C
Number: 15

Suppose

is finite-dimensional, with

. Prove that there exist 1-dimensional subspaces

of

such that

Page: 65
Section: 2.C
Number: 16

Suppose

are finite-dimensional subspaces of

such that

is a direct sum. Prove that

is finitedimensional and

[The exercise above deepens the analogy between direct sums of subspaces and disjoint unions of subsets. Specifically, compare this exercise to the following obvious statement: if a set is written as a disjoint union of finite subsets, then the number of elements in the set equals the sum of the numbers of elements in the disjoint subsets.]

Page: 65
Section: 2.C
Number: 17

You might guess, by analogy with the formula for the number of elements in the union of three subsets of a finite set, that if

are subspaces of a finite-dimensional vector space, then

Prove this or give a counterexample.

Page: 73
Section: 3.A
Number: 1

Suppose

. Define

by

Show that

is linear if and only if

.

Page: 73
Section: 3.A
Number: 2

Suppose

. Define

by

Show that

is linear if and only if

.

Page: 73
Section: 3.A
Number: 3

Suppose

. Show that there exist scalars

for

and

such that

for every

. [The exercise above shows that

has the form promised in the last item of Example 3.4.]

Page: 73
Section: 3.A
Number: 4

Suppose

and

is a list of vectors in

such that

is a linearly independent list in

. Prove that

is linearly independent.

Page: 73
Section: 3.A
Number: 5

Prove the assertion in 3.7.

Page: 73
Section: 3.A
Number: 6

Prove the assertions in 3.9.

Page: 73
Section: 3.A
Number: 58

CHAPTER

Linear Maps

Page: 73
Section: 3.A
Number: 7

Show that every linear map from a 1-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if

and

, then there exists

such that

for all

.

Page: 73
Section: 3.A
Number: 8

Give an example of a function

such that

for all

and all

but

is not linear. [The exercise above and the next exercise show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.]

Page: 73
Section: 3.A
Number: 9

Give an example of a function

such that

for all

but

is not linear. (Here

is thought of as a complex vector space.) [There also exists a function

such that

satisfies the additivity condition above but

is not linear. However, showing the existence of such a function involves considerably more advanced tools.]

Page: 73
Section: 3.A
Number: 10

Suppose

is a subspace of

with

. Suppose

and

(which means that

for some

). Define

by

Prove that

is not a linear map on

.

Page: 73
Section: 3.A
Number: 11

Suppose

is finite-dimensional. Prove that every linear map on a subspace of

can be extended to a linear map on

. In other words, show that if

is a subspace of

and

, then there exists

such that

for all

.

Page: 73
Section: 3.A
Number: 12

Suppose

is finite-dimensional with

, and suppose

is infinite-dimensional. Prove that

is infinite-dimensional.

Page: 73
Section: 3.A
Number: 13

Suppose

is a linearly dependent list of vectors in

. Suppose also that

. Prove that there exist

such that no

satisfies

for each

.

Page: 73
Section: 3.A
Number: 14

Suppose

is finite-dimensional with

. Prove that there exist

such that

.

Page: 83
Section: 3.B
Number: 1

Give an example of a linear map

such that

null

and dim range

.

Page: 83
Section: 3.B
Number: 2

Suppose

is a vector space and

are such that

Prove that

.

Page: 83
Section: 3.B
Number: 3

Suppose

is a list of vectors in

. Define

by

(a) What property of

corresponds to

spanning

? (b) What property of

corresponds to

being linearly independent?

Page: 83
Section: 3.B
Number: 4

Show that

is not a subspace of

.

Page: 83
Section: 3.B
Number: 5

Give an example of a linear map

such that

Page: 83
Section: 3.B
Number: 6

Prove that there does not exist a linear map

such that

Page: 83
Section: 3.B
Number: 7

Suppose

and

are finite-dimensional with

. Show that

is not injective

is not a subspace of

.

Page: 83
Section: 3.B
Number: 8

Suppose

and

are finite-dimensional with

. Show that

is not surjective

is not a subspace of

.

Page: 83
Section: 3.B
Number: 9

Suppose

is injective and

is linearly independent in

. Prove that

is linearly independent in

.

Page: 83
Section: 3.B
Number: 68

CHAPTER

Linear Maps

Page: 83
Section: 3.B
Number: 10

Suppose

spans

and

. Prove that the list

spans range

.

Page: 83
Section: 3.B
Number: 11

Suppose

are injective linear maps such that

makes sense. Prove that

is injective.

Page: 83
Section: 3.B
Number: 12

Suppose that

is finite-dimensional and that

. Prove that there exists a subspace

of

such that

null

and range

.

Page: 83
Section: 3.B
Number: 13

Suppose

is a linear map from

to

such that

Prove that

is surjective.

Page: 83
Section: 3.B
Number: 14

Suppose

is a 3-dimensional subspace of

and that

is a linear map from

to

such that null

. Prove that

is surjective.

Page: 83
Section: 3.B
Number: 15

Prove that there does not exist a linear map from

to

whose null space equals

Page: 83
Section: 3.B
Number: 16

Suppose there exists a linear map on

whose null space and range are both finite-dimensional. Prove that

is finite-dimensional.

Page: 83
Section: 3.B
Number: 17

Suppose

and

are both finite-dimensional. Prove that there exists an injective linear map from

to

if and only if

.

Page: 83
Section: 3.B
Number: 18

Suppose

and

are both finite-dimensional. Prove that there exists a surjective linear map from

onto

if and only if

.

Page: 83
Section: 3.B
Number: 19

Suppose

and

are finite-dimensional and that

is a subspace of

. Prove that there exists

such that null

if and only if

.

Page: 83
Section: 3.B
Number: 20

Suppose

is finite-dimensional and

. Prove that

is injective if and only if there exists

such that

is the identity map on

.

Page: 83
Section: 3.B
Number: 21

Suppose

is finite-dimensional and

. Prove that

is surjective if and only if there exists

such that

is the identity map on

. SECTION 3.B Null Spaces and Ranges

Page: 83
Section: 3.B
Number: 22

Suppose

and

are finite-dimensional vector spaces and

and

. Prove that

Page: 83
Section: 3.B
Number: 23

Suppose

and

are finite-dimensional vector spaces and

and

. Prove that

Page: 83
Section: 3.B
Number: 24

Suppose

is finite-dimensional and

. Prove that null

null

if and only if there exists

such that

.

Page: 83
Section: 3.B
Number: 25

Suppose

is finite-dimensional and

. Prove that range

range

if and only if there exists

such that

.

Page: 83
Section: 3.B
Number: 26

Suppose

is such that

for every nonconstant polynomial

. Prove that

is surjective. [The notation

is used above to remind you of the differentiation map that sends a polynomial

to

. Without knowing the formula for the derivative of a polynomial (except that it reduces the degree by 1), you can use the exercise above to show that for every polynomial

, there exists a polynomial

such that

.]

Page: 83
Section: 3.B
Number: 27

Suppose

. Prove that there exists a polynomial

such that

. [This exercise can be done without linear algebra, but it’s more fun to do it using linear algebra.]

Page: 83
Section: 3.B
Number: 28

Suppose

, and

is a basis of range

. Prove that there exist

such that

for every

.

Page: 83
Section: 3.B
Number: 29

Suppose

. Suppose

is not in null

. Prove that

Page: 83
Section: 3.B
Number: 30

Suppose

and

are linear maps from

to

that have the same null space. Show that there exists a constant

such that

.

Page: 83
Section: 3.B
Number: 31

Give an example of two linear maps

and

from

to

that have the same null space but are such that

is not a scalar multiple of

.

Page: 94
Section: 3.C
Number: 1

Suppose

and

are finite-dimensional and

. Show that with respect to each choice of bases of

and

, the matrix of

has at least dim range

nonzero entries.

Page: 94
Section: 3.C
Number: 2

Suppose

is the differentiation map defined by

. Find a basis of

and a basis of

such that the matrix of

with respect to these bases is

[Compare the exercise above to Example 3.34. The next exercise generalizes the exercise above.]

Page: 94
Section: 3.C
Number: 3

Suppose

and

are finite-dimensional and

. Prove that there exist a basis of

and a basis of

such that with respect to these bases, all entries of

are

except that the entries in row

, column

, equal

for

.

Page: 94
Section: 3.C
Number: 4

Suppose

is a basis of

and

is finite-dimensional. Suppose

. Prove that there exists a basis

of

such that all the entries in the first column of

(with respect to the bases

and

are

except for possibly a

in the first row, first column. [In this exercise, unlike Exercise 3, you are given the basis of

instead of being able to choose a basis of

.]

Page: 94
Section: 3.C
Number: 5

Suppose

is a basis of

and

is finite-dimensional. Suppose

. Prove that there exists a basis

of

such that all the entries in the first row of

(with respect to the bases

and

are

except for possibly a

in the first row, first column. [In this exercise, unlike Exercise 3, you are given the basis of

instead of being able to choose a basis of

.] SECTION 3.C Matrices

Page: 94
Section: 3.C
Number: 6

Suppose

and

are finite-dimensional and

. Prove that dim range

if and only if there exist a basis of

and a basis of

such that with respect to these bases, all entries of

equal

.

Page: 94
Section: 3.C
Number: 7

Verify

.

Page: 94
Section: 3.C
Number: 8

Verify

.

Page: 94
Section: 3.C
Number: 9

Prove 3.52.

Page: 94
Section: 3.C
Number: 10

Suppose

is an

-by-

matrix and

is an

-by-

matrix. Prove that

for

. In other words, show that row

of

equals (row

of

) times

.

Page: 94
Section: 3.C
Number: 11

Suppose

is a 1-by-

matrix and

is an

-by-

matrix. Prove that

In other words, show that

is a linear combination of the rows of

, with the scalars that multiply the rows coming from

.

Page: 94
Section: 3.C
Number: 12

Give an example with 2-by-2 matrices to show that matrix multiplication is not commutative. In other words, find 2-by-2 matrices

and

such that

.

Page: 94
Section: 3.C
Number: 13

Prove that the distributive property holds for matrix addition and matrix multiplication. In other words, suppose

, and

are matrices whose sizes are such that

and

make sense. Prove that

and

both make sense and that

and

.

Page: 94
Section: 3.C
Number: 14

Prove that matrix multiplication is associative. In other words, suppose

, and

are matrices whose sizes are such that

makes sense. Prove that

makes sense and that

.

Page: 94
Section: 3.C
Number: 15

Suppose

is an

-by-

matrix and

. Show that the entry in row

, column

, of

(which is defined to mean

) is

Page: 104
Section: 3.D
Number: 1

Suppose

and

are both invertible linear maps. Prove that

is invertible and that

.

Page: 104
Section: 3.D
Number: 2

Suppose

is finite-dimensional and

. Prove that the set of noninvertible operators on

is not a subspace of

.

Page: 104
Section: 3.D
Number: 3

Suppose

is finite-dimensional,

is a subspace of

, and

. Prove there exists an invertible operator

such that

for every

if and only if

is injective.

Page: 104
Section: 3.D
Number: 4

Suppose

is finite-dimensional and

. Prove that null

null

if and only if there exists an invertible operator

such that

.

Page: 104
Section: 3.D
Number: 5

Suppose

is finite-dimensional and

. Prove that range

range

if and only if there exists an invertible operator

such that

.

Page: 104
Section: 3.D
Number: 6

Suppose

and

are finite-dimensional and

. Prove that there exist invertible operators

and

such that

if and only if

null

.

Page: 104
Section: 3.D
Number: 7

Suppose

and

are finite-dimensional. Let

. Let

(a) Show that

is a subspace of

. (b) Suppose

. What is

? SECTION 3.D Invertibility and Isomorphic Vector Spaces

Page: 104
Section: 3.D
Number: 8

Suppose

is finite-dimensional and

is a surjective linear map of

onto

. Prove that there is a subspace

of

such that

is an isomorphism of

onto

. (Here

means the function

restricted to

. In other words,

is the function whose domain is

, with

defined by

for every

.)

Page: 104
Section: 3.D
Number: 9

Suppose

is finite-dimensional and

. Prove that

is invertible if and only if both

and

are invertible.

Page: 104
Section: 3.D
Number: 10

Suppose

is finite-dimensional and

. Prove that

if and only if

.

Page: 104
Section: 3.D
Number: 11

Suppose

is finite-dimensional and

and

. Show that

is invertible and that

.

Page: 104
Section: 3.D
Number: 12

Show that the result in the previous exercise can fail without the hypothesis that

is finite-dimensional.

Page: 104
Section: 3.D
Number: 13

Suppose

is a finite-dimensional vector space and

are such that

is surjective. Prove that

is injective.

Page: 104
Section: 3.D
Number: 14

Suppose

is a basis of

. Prove that the map

defined by

is an isomorphism of

onto

; here

is the matrix of

with respect to the basis

.

Page: 104
Section: 3.D
Number: 15

Prove that every linear map from

to

is given by a matrix multiplication. In other words, prove that if

, then there exists an

-by-

matrix

such that

for every

.

Page: 104
Section: 3.D
Number: 16

Suppose

is finite-dimensional and

. Prove that

is a scalar multiple of the identity if and only if

for every

.

Page: 104
Section: 3.D
Number: 17

Suppose

is finite-dimensional and

is a subspace of

such that

and

for all

and all

. Prove that

or

.

Page: 104
Section: 3.D
Number: 18

Show that

and

are isomorphic vector spaces.

Page: 104
Section: 3.D
Number: 19

Suppose

is such that

is injective and

for every nonzero polynomial

. (a) Prove that

is surjective. (b) Prove that

for every nonzero

.

Page: 104
Section: 3.D
Number: 90

CHAPTER

Linear Maps

Page: 104
Section: 3.D
Number: 20

Suppose

is a positive integer and

for

. Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables): (a) The trivial solution

is the only solution to the homogeneous system of equations

(b) For every

, there exists a solution to the system of equations

Page: 114
Section: 3.E
Number: 1

Suppose

is a function from

to

. The graph of

is the subset of

defined by

Prove that

is a linear map if and only if the graph of

is a subspace of

. [Formally, a function

from

to

is a subset

of

such that for each

, there exists exactly one element

. In other words, formally a function is what is called above its graph. We do not usually think of functions in this formal manner. However, if we do become formal, then the exercise above could be rephrased as follows: Prove that a function

from

to

is a linear map if and only if

is a subspace of

.] SECTION 3.E Products and Quotients of Vector Spaces

Page: 114
Section: 3.E
Number: 2

Suppose

are vector spaces such that

is finitedimensional. Prove that

is finite-dimensional for each

.

Page: 114
Section: 3.E
Number: 3

Give an example of a vector space

and subspaces

of

such that

is isomorphic to

but

is not a direct sum. Hint: The vector space

must be infinite-dimensional.

Page: 114
Section: 3.E
Number: 4

Suppose

are vector spaces. Prove that

and

are isomorphic vector spaces.

Page: 114
Section: 3.E
Number: 5

Suppose

are vector spaces. Prove that

and

are isomorphic vector spaces.

Page: 114
Section: 3.E
Number: 6

For

a positive integer, define

by

Prove that

and

are isomorphic vector spaces.

Page: 114
Section: 3.E
Number: 7

Suppose

are vectors in

and

are subspaces of

such that

. Prove that

.

Page: 114
Section: 3.E
Number: 8

Prove that a nonempty subset

of

is an affine subset of

if and only if

for all

and all

.

Page: 114
Section: 3.E
Number: 9

Suppose

and

are affine subsets of

. Prove that the intersection

is either an affine subset of

or the empty set.

Page: 114
Section: 3.E
Number: 10

Prove that the intersection of every collection of affine subsets of

is either an affine subset of

or the empty set.

Page: 114
Section: 3.E
Number: 11

Suppose

. Let

(a) Prove that

is an affine subset of

. (b) Prove that every affine subset of

that contains

also contains

. © Prove that

for some

and some subspace

of

with

.

Page: 114
Section: 3.E
Number: 12

Suppose

is a subspace of

such that

is finite-dimensional. Prove that

is isomorphic to

.

Page: 114
Section: 3.E
Number: 100

CHAPTER

Linear Maps

Page: 114
Section: 3.E
Number: 13

Suppose

is a subspace of

and

is a basis of

and

is a basis of

. Prove that

is a basis of

.

Page: 114
Section: 3.E
Number: 14

Suppose

for only finitely many

. (a) Show that

is a subspace of

. (b) Prove that

is infinite-dimensional.

Page: 114
Section: 3.E
Number: 15

Suppose

and

. Prove that

.

Page: 114
Section: 3.E
Number: 16

Suppose

is a subspace of

such that

. Prove that there exists

such that null

.

Page: 114
Section: 3.E
Number: 17

Suppose

is a subspace of

such that

is finite-dimensional. Prove that there exists a subspace

of

such that

and

.

Page: 114
Section: 3.E
Number: 18

Suppose

and

is a subspace of

. Let

denote the quotient map from

onto

. Prove that there exists

such that

if and only if

null

.

Page: 114
Section: 3.E
Number: 19

Find a correct statement analogous to

that is applicable to finite sets, with unions analogous to sums of subspaces and disjoint unions analogous to direct sums.

Page: 114
Section: 3.E
Number: 20

Suppose

is a subspace of

. Define

by

(a) Show that

is a linear map. (b) Show that

is injective. © Show that range

for every

.

Page: 129
Section: 3.F
Number: 1

Explain why every linear functional is either surjective or the zero map.

Page: 129
Section: 3.F
Number: 2

Give three distinct examples of linear functionals on

.

Page: 129
Section: 3.F
Number: 3

Suppose

is finite-dimensional and

with

. Prove that there exists

such that

.

Page: 129
Section: 3.F
Number: 4

Suppose

is finite-dimensional and

is a subspace of

such that

. Prove that there exists

such that

for every

but

.

Page: 129
Section: 3.F
Number: 5

Suppose

are vector spaces. Prove that

and

are isomorphic vector spaces.

Page: 129
Section: 3.F
Number: 6

Suppose

is finite-dimensional and

. Define a linear

by

(a) Prove that

spans

if and only if

is injective. (b) Prove that

is linearly independent if and only if

is surjective.

Suppose

is a positive integer. Show that the dual basis of the basis

of

is

, where

. Here

denotes the

derivative of

, with the understanding that the

derivative of

is

.

Page: 129
Section: 3.F
Number: 8

Suppose

is a positive integer. (a) Show that

is a basis of

. (b) What is the dual basis of the basis in part (a)?

Page: 129
Section: 3.F
Number: 9

Suppose

is a basis of

and

is the corresponding dual basis of

. Suppose

. Prove that

Page: 129
Section: 3.F
Number: 10

Prove the first two bullet points in 3.101.

Page: 129
Section: 3.F
Number: 114

CHAPTER

Linear Maps

Page: 129
Section: 3.F
Number: 11

Suppose

is an

-by-

matrix with

. Prove that the rank of

is

if and only if there exist

and

such that

for every

and every

.

Page: 129
Section: 3.F
Number: 12

Show that the dual map of the identity map on

is the identity map on

.

Page: 129
Section: 3.F
Number: 13

Define

by

. Suppose

denotes the dual basis of the standard basis of

and

denotes the dual basis of the standard basis of

. (a) Describe the linear functionals

and

. (b) Write

and

as linear combinations of

.

Page: 129
Section: 3.F
Number: 14

Define

by

for

. (a) Suppose

is defined by

. Describe the linear functional

on

. (b) Suppose

is defined by

. Evaluate

.

Page: 129
Section: 3.F
Number: 15

Suppose

is finite-dimensional and

. Prove that

if and only if

.

Page: 129
Section: 3.F
Number: 16

Suppose

and

are finite-dimensional. Prove that the map that takes

to

is an isomorphism of

onto

.

Page: 129
Section: 3.F
Number: 17

Suppose

. Explain why

null

.

Page: 129
Section: 3.F
Number: 18

Suppose

is finite-dimensional and

. Show that

if and only if

.

Page: 129
Section: 3.F
Number: 19

Suppose

is finite-dimensional and

is a subspace of

. Show that

if and only if

.

Page: 129
Section: 3.F
Number: 20

Suppose

and

are subsets of

with

. Prove that

.

Page: 129
Section: 3.F
Number: 21

Suppose

is finite-dimensional and

and

are subspaces of

with

. Prove that

.

Page: 129
Section: 3.F
Number: 22

Suppose

are subspaces of

. Show that

. SECTION 3.F Duality

Page: 129
Section: 3.F
Number: 23

Suppose

is finite-dimensional and

and

are subspaces of

. Prove that

.

Page: 129
Section: 3.F
Number: 24

Prove

using the ideas sketched in the discussion before the statement of

.

Page: 129
Section: 3.F
Number: 25

Suppose

is finite-dimensional and

is a subspace of

. Show that

Page: 129
Section: 3.F
Number: 26

Suppose

is finite-dimensional and

is a subspace of

. Show that

Page: 129
Section: 3.F
Number: 27

Suppose

and null

, where

is the linear functional on

defined by

. Prove that range

.

Page: 129
Section: 3.F
Number: 28

Suppose

and

are finite-dimensional,

, and there exists

such that null

. Prove that range

.

Page: 129
Section: 3.F
Number: 29

Suppose

and

are finite-dimensional,

, and there exists

such that range

. Prove that null

.

Page: 129
Section: 3.F
Number: 30

Suppose

is finite-dimensional and

is a linearly independent list in

. Prove that

Page: 129
Section: 3.F
Number: 31

Suppose

is finite-dimensional and

is a basis of

. Show that there exists a basis of

whose dual basis is

.

Page: 129
Section: 3.F
Number: 32

Suppose

, and

and

are bases of

. Prove that the following are equivalent: (a)

is invertible. (b) The columns of

are linearly independent in

. © The columns of

. (d) The rows of

are linearly independent in

. (e) The rows of

. Here

means

.

Page: 129
Section: 3.F
Number: 116

CHAPTER

Linear Maps

Page: 129
Section: 3.F
Number: 33

Suppose

and

are positive integers. Prove that the function that takes

to

is a linear map from

to

. Furthermore, prove that this linear map is invertible.

Page: 129
Section: 3.F
Number: 34

The double dual space of

, denoted

, is defined to be the dual space of

. In other words,

. Define

by

for

and

. (a) Show that

is a linear map from

to

. (b) Show that if

, then

, where

. © Show that if

is finite-dimensional, then

is an isomorphism from

onto

. [Suppose

is finite-dimensional. Then

and

are isomorphic, but finding an isomorphism from

onto

generally requires choosing a basis of

. In contrast, the isomorphism

from

onto

does not require a choice of basis and thus is considered more natural.]

Page: 129
Section: 3.F
Number: 35

Show that

and

are isomorphic.

Page: 129
Section: 3.F
Number: 36

Suppose

is a subspace of

. Let

be the inclusion map defined by

. Thus

. (a) Show that null

. (b) Prove that if

is finite-dimensional, then range

. © Prove that if

is finite-dimensional, then

is an isomorphism from

onto

. [The isomorphism in part © is natural in that it does not depend on a choice of basis in either vector space.]

Page: 129
Section: 3.F
Number: 37

Suppose

is a subspace of

. Let

be the usual quotient map. Thus

. (a) Show that

is injective. (b) Show that range

. © Conclude that

is an isomorphism from

onto

. [The isomorphism in part © is natural in that it does not depend on a choice of basis in either vector space. In fact, there is no assumption here that any of these vector spaces are finite-dimensional.]

Page: 145
Section: 4
Number: 1

Verify all the assertions in

except the last one.

Page: 145
Section: 4
Number: 2

Suppose

is a positive integer. Is the set

a subspace of

?

Page: 145
Section: 4
Number: 3

Is the set

a subspace of

?

Page: 145
Section: 4
Number: 4

Suppose

and

are positive integers with

, and suppose

. Prove that there exists a polynomial

with

such that

and such that

has no other zeros.

Page: 145
Section: 4
Number: 5

Suppose

is a nonnegative integer,

are distinct elements of

, and

. Prove that there exists a unique polynomial

such that

for

. [This result can be proved without using linear algebra. However, try to find the clearer, shorter proof that uses some linear algebra.]

Page: 145
Section: 4
Number: 6

Suppose

has degree

. Prove that

has

distinct zeros if and only if

and its derivative

have no zeros in common.

Page: 145
Section: 4
Number: 7

Prove that every polynomial of odd degree with real coefficients has a real zero.

Page: 145
Section: 4
Number: 8

Define

by

Show that

for every polynomial

and that

is a linear map.

Page: 145
Section: 4
Number: 130

CHAPTER

Polynomials

Page: 145
Section: 4
Number: 9

Suppose

. Define

by

Prove that

is a polynomial with real coefficients.

Page: 145
Section: 4
Number: 10

Suppose

is a nonnegative integer and

is such that there exist distinct real numbers

such that

for

. Prove that all the coefficients of

are real.

Page: 145
Section: 4
Number: 11

Suppose

with

. Let

. (a) Show that

. (b) Find a basis of

.

Page: 154
Section: 5.A
Number: 1

Suppose

and

is a subspace of

. (a) Prove that if

null

, then

is invariant under

. (b) Prove that if range

, then

is invariant under

.

Page: 154
Section: 5.A
Number: 2

Suppose

are such that

. Prove that null

is invariant under

. SECTION 5.A Invariant Subspaces

Page: 154
Section: 5.A
Number: 3

Suppose

are such that

. Prove that range

is invariant under

.

Page: 154
Section: 5.A
Number: 4

Suppose that

and

are subspaces of

invariant under

. Prove that

is invariant under

.

Page: 154
Section: 5.A
Number: 5

Suppose

. Prove that the intersection of every collection of subspaces of

invariant under

is invariant under

.

Page: 154
Section: 5.A
Number: 6

Prove or give a counterexample: if

is finite-dimensional and

is a subspace of

that is invariant under every operator on

, then

or

.

Page: 154
Section: 5.A
Number: 7

Suppose

is defined by

. Find the eigenvalues of

.

Page: 154
Section: 5.A
Number: 8

Define

by

Find all eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 9

Define

by

Find all eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 10

Define

by

(a) Find all eigenvalues and eigenvectors of

. (b) Find all invariant subspaces of

.

Page: 154
Section: 5.A
Number: 11

Define

by

. Find all eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 12

Define

by

for all

. Find all eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 13

Suppose

is finite-dimensional,

, and

. Prove that there exists

such that

and

is invertible.

Page: 154
Section: 5.A
Number: 140

CHAPTER

Eigenvalues, Eigenvectors, and Invariant Subspaces

Page: 154
Section: 5.A
Number: 14

Suppose

, where

and

are nonzero subspaces of

. Define

by

for

and

. Find all eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 15

Suppose

. Suppose

is invertible. (a) Prove that

and

have the same eigenvalues. (b) What is the relationship between the eigenvectors of

and the eigenvectors of

?

Page: 154
Section: 5.A
Number: 16

Suppose

is a complex vector space,

, and the matrix of

with respect to some basis of

contains only real entries. Show that if

is an eigenvalue of

, then so is

.

Page: 154
Section: 5.A
Number: 17

Give an example of an operator

such that

has no (real) eigenvalues.

Page: 154
Section: 5.A
Number: 18

Show that the operator

defined by

has no eigenvalues.

Page: 154
Section: 5.A
Number: 19

Suppose

is a positive integer and

is defined by

in other words,

is the operator whose matrix (with respect to the standard basis) consists of all 1’s. Find all eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 20

Find all eigenvalues and eigenvectors of the backward shift operator

defined by

Page: 154
Section: 5.A
Number: 21

Suppose

is invertible. (a) Suppose

with

. Prove that

is an eigenvalue of

if and only if

is an eigenvalue of

. (b) Prove that

and

have the same eigenvectors. SECTION 5.A Invariant Subspaces

Page: 154
Section: 5.A
Number: 22

Suppose

and there exist nonzero vectors

and

in

such that

Prove that

or

is an eigenvalue of

.

Page: 154
Section: 5.A
Number: 23

Suppose

is finite-dimensional and

. Prove that

and

have the same eigenvalues.

Page: 154
Section: 5.A
Number: 24

Suppose

is an

-by-

matrix with entries in

. Define

by

, where elements of

are thought of as

-by-

column vectors. (a) Suppose the sum of the entries in each row of

equals

. Prove that

is an eigenvalue of

. (b) Suppose the sum of the entries in each column of

equals

. Prove that

is an eigenvalue of

.

Page: 154
Section: 5.A
Number: 25

Suppose

and

are eigenvectors of

such that

is also an eigenvector of

. Prove that

and

are eigenvectors of

corresponding to the same eigenvalue.

Page: 154
Section: 5.A
Number: 26

Suppose

is such that every nonzero vector in

is an eigenvector of

. Prove that

is a scalar multiple of the identity operator.

Page: 154
Section: 5.A
Number: 27

Suppose

is finite-dimensional and

is such that every subspace of

with dimension

is invariant under

. Prove that

is a scalar multiple of the identity operator.

Page: 154
Section: 5.A
Number: 28

Suppose

is finite-dimensional with

and

is such that every 2-dimensional subspace of

is invariant under

. Prove that

is a scalar multiple of the identity operator.

Page: 154
Section: 5.A
Number: 29

Suppose

and dim range

. Prove that

has at most

distinct eigenvalues.

Page: 154
Section: 5.A
Number: 30

Suppose

and

, and

are eigenvalues of

. Prove that there exists

such that

.

Page: 154
Section: 5.A
Number: 31

Suppose

is finite-dimensional and

is a list of vectors in

. Prove that

is linearly independent if and only if there exists

such that

are eigenvectors of

corresponding to distinct eigenvalues.

Page: 154
Section: 5.A
Number: 142

CHAPTER

Eigenvalues, Eigenvectors, and Invariant Subspaces

Page: 154
Section: 5.A
Number: 32

Suppose

is a list of distinct real numbers. Prove that the list

is linearly independent in the vector space of real-valued functions on

. Hint: Let

, and define an operator

by

. Find eigenvalues and eigenvectors of

.

Page: 154
Section: 5.A
Number: 33

Suppose

. Prove that

range

.

Page: 154
Section: 5.A
Number: 34

Suppose

. Prove that

null

) is injective if and only if (null

range

.

Page: 154
Section: 5.A
Number: 35

Suppose

is finite-dimensional,

, and

is invariant under

. Prove that each eigenvalue of

is an eigenvalue of

. [The exercise below asks you to verify that the hypothesis that

is finite-dimensional is needed for the exercise above.]

Page: 154
Section: 5.A
Number: 36

Give an example of a vector space

, an operator

, and a subspace

of

that is invariant under

such that

has an eigenvalue that is not an eigenvalue of

.

Page: 169
Section: 5.B
Number: 1

Suppose

and there exists a positive integer

such that

. (a) Prove that

is invertible and that

(b) Explain how you would guess the formula above.

Page: 169
Section: 5.B
Number: 2

Suppose

and

. Suppose

is an eigenvalue of

. Prove that

or

.

Page: 169
Section: 5.B
Number: 3

Suppose

and

is not an eigenvalue of

. Prove that

.

Page: 169
Section: 5.B
Number: 4

Suppose

and

. Prove that

null

range

.

Page: 169
Section: 5.B
Number: 5

Suppose

and

is invertible. Suppose

is a polynomial. Prove that

Page: 169
Section: 5.B
Number: 6

Suppose

and

is a subspace of

invariant under

. Prove that

is invariant under

for every polynomial

.

Page: 169
Section: 5.B
Number: 7

Suppose

. Prove that

is an eigenvalue of

if and only if

or

is an eigenvalue of

.

Page: 169
Section: 5.B
Number: 8

Give an example of

such that

.

Page: 169
Section: 5.B
Number: 9

Suppose

is finite-dimensional,

, and

with

. Let

be a nonzero polynomial of smallest degree such that

. Prove that every zero of

is an eigenvalue of

.

Page: 169
Section: 5.B
Number: 10

Suppose

and

is an eigenvector of

with eigenvalue

. Suppose

. Prove that

.

Page: 169
Section: 5.B
Number: 11

Suppose

is a polynomial, and

. Prove that

is an eigenvalue of

if and only if

for some eigenvalue

of

.

Page: 169
Section: 5.B
Number: 12

Show that the result in the previous exercise does not hold if

is replaced with

.

Page: 169
Section: 5.B
Number: 154

CHAPTER

Eigenvalues, Eigenvectors, and Invariant Subspaces

Page: 169
Section: 5.B
Number: 13

Suppose

is a complex vector space and

has no eigenvalues. Prove that every subspace of

invariant under

is either

or infinitedimensional.

Page: 169
Section: 5.B
Number: 14

Give an example of an operator whose matrix with respect to some basis contains only 0’s on the diagonal, but the operator is invertible. [The exercise above and the exercise below show that

fails without the hypothesis that an upper-triangular matrix is under consideration.]

Page: 169
Section: 5.B
Number: 15

Give an example of an operator whose matrix with respect to some basis contains only nonzero numbers on the diagonal, but the operator is not invertible.

Page: 169
Section: 5.B
Number: 16

Rewrite the proof of

using the linear map that sends

to

(and use 3.23).

Page: 169
Section: 5.B
Number: 17

Rewrite the proof of

using the linear map that sends

to

(and use 3.23).

Page: 169
Section: 5.B
Number: 18

Suppose

is a finite-dimensional complex vector space and

. Define a function

by

Prove that

is not a continuous function.

Page: 169
Section: 5.B
Number: 19

Suppose

is finite-dimensional with

and

. Prove that

Page: 169
Section: 5.B
Number: 20

Suppose

is a finite-dimensional complex vector space and

. Prove that

has an invariant subspace of dimension

for each

.

Page: 176
Section: 5.C
Number: 1

Suppose

is diagonalizable. Prove that

null

range

.

Page: 176
Section: 5.C
Number: 2

Prove the converse of the statement in the exercise above or give a counterexample to the converse.

Page: 176
Section: 5.C
Number: 3

Suppose

is finite-dimensional and

. Prove that the following are equivalent: (a)

. (b)

. © null

range

.

Page: 176
Section: 5.C
Number: 4

Give an example to show that the exercise above is false without the hypothesis that

is finite-dimensional.

Page: 176
Section: 5.C
Number: 5

Suppose

is a finite-dimensional complex vector space and

. Prove that

is diagonalizable if and only if

for every

.

Page: 176
Section: 5.C
Number: 6

Suppose

is finite-dimensional,

has

distinct eigenvalues, and

has the same eigenvectors as

(not necessarily with the same eigenvalues). Prove that

.

Page: 176
Section: 5.C
Number: 7

Suppose

has a diagonal matrix

with respect to some basis of

and that

. Prove that

appears on the diagonal of

precisely

times.

Page: 176
Section: 5.C
Number: 8

Suppose

and

. Prove that

or

is invertible. SECTION 5.C Eigenspaces and Diagonal Matrices

Page: 176
Section: 5.C
Number: 9

Suppose

is invertible. Prove that

for every

with

.

Page: 176
Section: 5.C
Number: 10

Suppose that

is finite-dimensional and

. Let

denote the distinct nonzero eigenvalues of

. Prove that

Page: 176
Section: 5.C
Number: 11

Verify the assertion in Example 5.40.

Page: 176
Section: 5.C
Number: 12

Suppose

each have

as eigenvalues. Prove that there exists an invertible operator

such that

.

Page: 176
Section: 5.C
Number: 13

Find

such that

and

each have

as eigenvalues,

and

have no other eigenvalues, and there does not exist an invertible operator

such that

.

Page: 176
Section: 5.C
Number: 14

Find

such that

and

are eigenvalues of

and such that

does not have a diagonal matrix with respect to any basis of

.

Page: 176
Section: 5.C
Number: 15

Suppose

is such that

and

are eigenvalues of

. Furthermore, suppose

does not have a diagonal matrix with respect to any basis of

. Prove that there exists

such that

.

Page: 176
Section: 5.C
Number: 16

The Fibonacci sequence

is defined by

Define

by

. (a) Show that

for each positive integer

. (b) Find the eigenvalues of

. © Find a basis of

consisting of eigenvectors of

. (d) Use the solution to part © to compute

. Conclude that

for each positive integer

. (e) Use part (d) to conclude that for each positive integer

, the Fibonacci number

is the integer that is closest to

Page: 190
Section: 6.A
Number: 1

Show that the function that takes

to

is not an inner product on

.

Page: 190
Section: 6.A
Number: 2

Show that the function that takes

to

is not an inner product on

.

Page: 190
Section: 6.A
Number: 3

Suppose

and

. Replace the positivity condition (which states that

for all

) in the definition of an inner product (6.3) with the condition that

for some

. Show that this change in the definition does not change the set of functions from

to

that are inner products on

.

Page: 190
Section: 6.A
Number: 4

Suppose

is a real inner product space. (a) Show that

for every

. (b) Show that if

have the same norm, then

is orthogonal to

. © Use part (b) to show that the diagonals of a rhombus are perpendicular to each other.

Page: 190
Section: 6.A
Number: 5

Suppose

is such that

for every

. Prove that

is invertible.

Page: 190
Section: 6.A
Number: 6

Suppose

. Prove that

if and only if

for all

.

Page: 190
Section: 6.A
Number: 7

Suppose

. Prove that

for all

if and only if

.

Page: 190
Section: 6.A
Number: 8

Suppose

and

. Prove that

.

Page: 190
Section: 6.A
Number: 9

Suppose

and

. Prove that

Page: 190
Section: 6.A
Number: 10

Find vectors

such that

is a scalar multiple of

is orthogonal to

, and

.

Page: 190
Section: 6.A
Number: 176

CHAPTER

Inner Product Spaces

Page: 190
Section: 6.A
Number: 11

Prove that

for all positive numbers

.

Page: 190
Section: 6.A
Number: 12

Prove that

for all positive integers

and all real numbers

.

Page: 190
Section: 6.A
Number: 13

Suppose

are nonzero vectors in

. Prove that

where

is the angle between

and

(thinking of

and

as arrows with initial point at the origin). Hint: Draw the triangle formed by

, and

; then use the law of cosines.

Page: 190
Section: 6.A
Number: 14

The angle between two vectors (thought of as arrows with initial point at the origin) in

or

can be defined geometrically. However, geometry is not as clear in

for

. Thus the angle between two nonzero vectors

is defined to be

where the motivation for this definition comes from the previous exercise. Explain why the Cauchy-Schwarz Inequality is needed to show that this definition makes sense.

Page: 190
Section: 6.A
Number: 15

Prove that

for all real numbers

and

.

Page: 190
Section: 6.A
Number: 16

Suppose

are such that

What number does

equal? SECTION 6.A Inner Products and Norms

Page: 190
Section: 6.A
Number: 17

Prove or disprove: there is an inner product on

such that the associated norm is given by

for all

.

Page: 190
Section: 6.A
Number: 18

Suppose

. Prove that there is an inner product on

such that the associated norm is given by

for all

if and only if

.

Page: 190
Section: 6.A
Number: 19

Suppose

is a real inner product space. Prove that

for all

.

Page: 190
Section: 6.A
Number: 20

Suppose

is a complex inner product space. Prove that

for all

.

Page: 190
Section: 6.A
Number: 21

A norm on a vector space

is a function ||

such that

if and only if

for all

and all

, and

for all

. Prove that a norm satisfying the parallelogram equality comes from an inner product (in other words, show that if || is a norm on

satisfying the parallelogram equality, then there is an inner product

,

such that

for all

.

Page: 190
Section: 6.A
Number: 22

Show that the square of an average is less than or equal to the average of the squares. More precisely, show that if

, then the square of the average of

is less than or equal to the average of

.

Page: 190
Section: 6.A
Number: 23

Suppose

are inner product spaces. Show that the equation

defines an inner product on

. [In the expression above on the right,

denotes the inner product on

. Each of the spaces

may have a different inner product, even though the same notation is used here.]

Page: 190
Section: 6.A
Number: 178

CHAPTER

Inner Product Spaces

Page: 190
Section: 6.A
Number: 24

Suppose

is an injective operator on

. Define

by

for

. Show that

is an inner product on

.

Page: 190
Section: 6.A
Number: 25

Suppose

is not injective. Define

as in the exercise above. Explain why

is not an inner product on

.

Page: 190
Section: 6.A
Number: 26

Suppose

are differentiable functions from

to

. (a) Show that

(b) Suppose

and

for every

. Show that

for every

. © Interpret the result in part (b) geometrically in terms of the tangent vector to a curve lying on a sphere in

centered at the origin. [For the exercise above, a function

is called differentiable if there exist differentiable functions

from

to

such that

for each

. Furthermore, for each

, the derivative

is defined by

.]

Page: 190
Section: 6.A
Number: 27

Suppose

. Prove that

Page: 190
Section: 6.A
Number: 28

Suppose

is a subset of

with the property that

implies

. Let

. Show that there is at most one point in

that is closest to

. In other words, show that there is at most one

such that

Hint: Use the previous exercise.

Page: 190
Section: 6.A
Number: 29

For

, define

. (a) Show that

is a metric on

. (b) Show that if

is finite-dimensional, then

is a complete metric on

(meaning that every Cauchy sequence converges). © Show that every finite-dimensional subspace of

is a closed subset of

(with respect to the metric

). SECTION 6.A Inner Products and Norms

Page: 190
Section: 6.A
Number: 30

Fix a positive integer

. The Laplacian

of a twice differentiable function

on

is the function on

defined by

The function

is called harmonic if

. A polynomial on

is a linear combination of functions of the form

, where

are nonnegative integers. Suppose

is a polynomial on

. Prove that there exists a harmonic polynomial

on

such that

for every

with

. [The only fact about harmonic functions that you need for this exercise is that if

is a harmonic function on

and

for all

with

, then

.] Hint: A reasonable guess is that the desired harmonic polynomial

is of the form

for some polynomial

. Prove that there is a polynomial

on

such that

is harmonic by defining an operator

on a suitable vector space by

and then showing that

is injective and hence surjective.

Page: 190
Section: 6.A
Number: 31

Use inner products to prove Apollonius’s Identity: In a triangle with sides of length

, and

, let

be the length of the line segment from the midpoint of the side of length

to the opposite vertex. Then

Page: 204
Section: 6.B
Number: 1

(a) Suppose

. Show that

and

are orthonormal bases of

. (b) Show that each orthonormal basis of

is of the form given by one of the two possibilities of part (a).

Page: 204
Section: 6.B
Number: 2

Suppose

is an orthonormal list of vectors in

. Let

. Prove that

if and only if

.

Page: 204
Section: 6.B
Number: 3

Suppose

has an upper-triangular matrix with respect to the basis

. Find an orthonormal basis of

(use the usual inner product on

) with respect to which

has an upper-triangular matrix.

Page: 204
Section: 6.B
Number: 190

CHAPTER

Inner Product Spaces

Page: 204
Section: 6.B
Number: 4

Suppose

is a positive integer. Prove that

is an orthonormal list of vectors in

, the vector space of continuous real-valued functions on

with inner product

[The orthonormal list above is often used for modeling periodic phenomena such as tides.]

Page: 204
Section: 6.B
Number: 5

On

, consider the inner product given by

Apply the Gram-Schmidt Procedure to the basis

to produce an orthonormal basis of

.

Page: 204
Section: 6.B
Number: 6

Find an orthonormal basis of

(with inner product as in Exercise 5) such that the differentiation operator (the operator that takes

to

) on

has an upper-triangular matrix with respect to this basis.

Page: 204
Section: 6.B
Number: 7

Find a polynomial

such that

for every

.

Page: 204
Section: 6.B
Number: 8

Find a polynomial

such that

for every

.

Page: 204
Section: 6.B
Number: 9

What happens if the Gram-Schmidt Procedure is applied to a list of vectors that is not linearly independent? SECTION 6.B Orthonormal Bases

Page: 204
Section: 6.B
Number: 10

Suppose

is a real inner product space and

is a linearly independent list of vectors in

. Prove that there exist exactly

orthonormal lists

of vectors in

such that

for all

.

Page: 204
Section: 6.B
Number: 11

Suppose

and

are inner products on

such that

if and only if

. Prove that there is a positive number

such that

for every

.

Page: 204
Section: 6.B
Number: 12

Suppose

is finite-dimensional and

are inner products on

with corresponding norms

and

. Prove that there exists a positive number

such that

for every

.

Page: 204
Section: 6.B
Number: 13

Suppose

is a linearly independent list in

. Show that there exists

such that

for all

.

Page: 204
Section: 6.B
Number: 14

Suppose

is an orthonormal basis of

and

are vectors in

such that

for each

. Prove that

is a basis of

.

Page: 204
Section: 6.B
Number: 15

Suppose

is the vector space of continuous real-valued functions on the interval

with inner product given by

for

. Let

be the linear functional on

defined by

. Show that there does not exist

such that

for every

. [The exercise above shows that the Riesz Representation Theorem (6.42) does not hold on infinite-dimensional vector spaces without additional hypotheses on

and

.]

Page: 204
Section: 6.B
Number: 192

CHAPTER

Inner Product Spaces

Page: 204
Section: 6.B
Number: 16

Suppose

is finite-dimensional,

, all the eigenvalues of

have absolute value less than

, and

. Prove that there exists a positive integer

such that

for every

.

Page: 204
Section: 6.B
Number: 17

For

, let

denote the linear functional on

defined by

for

. (a) Show that if

, then

is a linear map from

to

. (Recall from Section 3.F that

and that

is called the dual space of

.) (b) Show that if

and

, then

is not a linear map. © Show that

is injective. (d) Suppose

and

is finite-dimensional. Use parts (a) and © and a dimension-counting argument (but without using 6.42) to show that

is an isomorphism from

onto

. [Part (d) gives an alternative proof of the Riesz Representation Theorem (6.42) when

. Part (d) also gives a natural isomorphism (meaning that it does not depend on a choice of basis) from a finite-dimensional real inner product space onto its dual space.]

Page: 216
Section: 6.C
Number: 1

Suppose

. Prove that

Page: 216
Section: 6.C
Number: 2

Suppose

is a finite-dimensional subspace of

. Prove that

if and only if

. [Exercise 14(a) shows that the result above is not true without the hypothesis that

is finite-dimensional.]

Page: 216
Section: 6.C
Number: 3

Suppose

is a subspace of

with basis

and

is a basis of

. Prove that if the Gram-Schmidt Procedure is applied to the basis of

above, producing a list

, then

is an orthonormal basis of

and

is an orthonormal basis of

.

Page: 216
Section: 6.C
Number: 4

Suppose

is the subspace of

defined by

Find an orthonormal basis of

and an orthonormal basis of

.

Page: 216
Section: 6.C
Number: 5

Suppose

is finite-dimensional and

is a subspace of

. Show that

, where

is the identity operator on

.

Page: 216
Section: 6.C
Number: 6

Suppose

and

are finite-dimensional subspaces of

. Prove that

if and only if

for all

and all

.

Page: 216
Section: 6.C
Number: 7

Suppose

is finite-dimensional and

is such that

and every vector in null

is orthogonal to every vector in range

. Prove that there exists a subspace

of

such that

.

Page: 216
Section: 6.C
Number: 8

Suppose

is finite-dimensional and

is such that

and

for every

. Prove that there exists a subspace

of

such that

.

Page: 216
Section: 6.C
Number: 9

Suppose

and

is a finite-dimensional subspace of

. Prove that

is invariant under

if and only if

.

Page: 216
Section: 6.C
Number: 202

CHAPTER

Inner Product Spaces

Page: 216
Section: 6.C
Number: 10

Suppose

is finite-dimensional,

, and

is a subspace of

. Prove that

and

are both invariant under

if and only if

.

Page: 216
Section: 6.C
Number: 11

In

, let

Find

such that

is as small as possible.

Page: 216
Section: 6.C
Number: 12

Find

such that

, and

is as small as possible.

Page: 216
Section: 6.C
Number: 13

Find

that makes

as small as possible. [The polynomial

is an excellent approximation to the answer to this exercise, but here you are asked to find the exact solution, which involves powers of

. A computer that can perform symbolic integration will be useful.]

Page: 216
Section: 6.C
Number: 14

Suppose

is the vector space of continuous real-valued functions on the interval

with inner product given by

for

. Let

be the subspace of

defined by

(a) Show that

. (b) Show that

and

do not hold without the finite-dimensional hypothesis.

Page: 229
Section: 7.A
Number: 1

Suppose

is a positive integer. Define

by

Find a formula for

.

Page: 229
Section: 7.A
Number: 2

Suppose

and

. Prove that

is an eigenvalue of

if and only if

is an eigenvalue of

.

Page: 229
Section: 7.A
Number: 3

Suppose

and

is a subspace of

. Prove that

is invariant under

if and only if

is invariant under

.

Page: 229
Section: 7.A
Number: 4

Suppose

. Prove that (a)

is injective if and only if

is surjective; (b)

is surjective if and only if

is injective.

Page: 229
Section: 7.A
Number: 5

Prove that

and

for every

. SECTION 7.A Self-Adjoint and Normal Operators

Page: 229
Section: 7.A
Number: 6

Make

into an inner product space by defining

Define

by

. (a) Show that

is not self-adjoint. (b) The matrix of

with respect to the basis

is

This matrix equals its conjugate transpose, even though

is not self-adjoint. Explain why this is not a contradiction.

Page: 229
Section: 7.A
Number: 7

Suppose

are self-adjoint. Prove that

is self-adjoint if and only if

.

Page: 229
Section: 7.A
Number: 8

Suppose

is a real inner product space. Show that the set of self-adjoint operators on

is a subspace of

.

Page: 229
Section: 7.A
Number: 9

Suppose

is a complex inner product space with

. Show that the set of self-adjoint operators on

is not a subspace of

.

Page: 229
Section: 7.A
Number: 10

Suppose

. Show that the set of normal operators on

is not a subspace of

.

Page: 229
Section: 7.A
Number: 11

Suppose

is such that

. Prove that there is a subspace

of

such that

if and only if

is self-adjoint.

Page: 229
Section: 7.A
Number: 12

Suppose that

is a normal operator on

and that

and

are eigenvalues of

. Prove that there exists a vector

such that

and

.

Page: 229
Section: 7.A
Number: 13

Give an example of an operator

such that

is normal but not self-adjoint.

Page: 229
Section: 7.A
Number: 14

Suppose

is a normal operator on

. Suppose also that

satisfy the equations

Show that

.

Page: 229
Section: 7.A
Number: 216

CHAPTER

Operators on Inner Product Spaces

Page: 229
Section: 7.A
Number: 15

Fix

. Define

by

for every

. (a) Suppose

. Prove that

is self-adjoint if and only if

is linearly dependent. (b) Prove that

is normal if and only if

is linearly dependent.

Page: 229
Section: 7.A
Number: 16

Suppose

is normal. Prove that

Page: 229
Section: 7.A
Number: 17

Suppose

is normal. Prove that

for every positive integer

.

Page: 229
Section: 7.A
Number: 18

Prove or give a counterexample: If

and there exists an orthonormal basis

of

such that

for each

, then

is normal.

Page: 229
Section: 7.A
Number: 19

Suppose

is normal and

. Suppose

. Prove that

.

Page: 229
Section: 7.A
Number: 20

Suppose

and

. Let

be the isomorphism from

onto the dual space

given by Exercise

in Section 6.B, and let

be the corresponding isomorphism from

onto

. Show that if

and

are used to identify

and

with

and

, then

is identified with the dual map

. More precisely, show that

.

Page: 229
Section: 7.A
Number: 21

Fix a positive integer

. In the inner product space of continuous realvalued functions on

with inner product

let

(a) Define

by

. Show that

. Conclude that

is normal but not self-adjoint. (b) Define

by

. Show that

is self-adjoint.

Page: 238
Section: 7.B
Number: 1

True or false (and give a proof of your answer): There exists

such that

is not self-adjoint (with respect to the usual inner product) and such that there is a basis of

consisting of eigenvectors of

.

Page: 238
Section: 7.B
Number: 2

Suppose that

is a self-adjoint operator on a finite-dimensional inner product space and that

and

are the only eigenvalues of

. Prove that

.

Page: 238
Section: 7.B
Number: 3

Give an example of an operator

such that

and

are the only eigenvalues of

and

.

Page: 238
Section: 7.B
Number: 4

Suppose

and

. Prove that

is normal if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of

are orthogonal and

where

denote the distinct eigenvalues of

.

Page: 238
Section: 7.B
Number: 5

Suppose

and

. Prove that

is self-adjoint if and only if all pairs of eigenvectors corresponding to distinct eigenvalues of

are orthogonal and

where

denote the distinct eigenvalues of

.

Page: 238
Section: 7.B
Number: 6

Prove that a normal operator on a complex inner product space is selfadjoint if and only if all its eigenvalues are real. [The exercise above strengthens the analogy (for normal operators) between self-adjoint operators and real numbers.]

Page: 238
Section: 7.B
Number: 7

Suppose

is a complex inner product space and

is a normal operator such that

. Prove that

is self-adjoint and

.

Page: 238
Section: 7.B
Number: 8

Give an example of an operator

on a complex vector space such that

but

.

Page: 238
Section: 7.B
Number: 9

Suppose

is a complex inner product space. Prove that every normal operator on

has a square root. (An operator

is called a square root of

if

.)

Page: 238
Section: 7.B
Number: 224

CHAPTER

Operators on Inner Product Spaces

Page: 238
Section: 7.B
Number: 10

Give an example of a real inner product space

and

and real numbers

with

such that

is not invertible. [The exercise above shows that the hypothesis that

is self-adjoint is needed in 7.26, even for real vector spaces.]

Page: 238
Section: 7.B
Number: 11

Prove or give a counterexample: every self-adjoint operator on

has a cube root. (An operator

is called a cube root of

if

.)

Page: 238
Section: 7.B
Number: 12

Suppose

is self-adjoint,

, and

. Suppose there exists

such that

and

Prove that

has an eigenvalue

such that

.

Page: 238
Section: 7.B
Number: 13

Give an alternative proof of the Complex Spectral Theorem that avoids Schur’s Theorem and instead follows the pattern of the proof of the Real Spectral Theorem.

Page: 238
Section: 7.B
Number: 14

Suppose

is a finite-dimensional real vector space and

. Prove that

has a basis consisting of eigenvectors of

if and only if there is an inner product on

that makes

into a self-adjoint operator.

Page: 238
Section: 7.B
Number: 15

Find the matrix entry below that is covered up.

Page: 246
Section: 7.C
Number: 1

Prove or give a counterexample: If

is self-adjoint and there exists an orthonormal basis

of

such that

for each

, then

is a positive operator.

Page: 246
Section: 7.C
Number: 2

Suppose

is a positive operator on

. Suppose

are such that

Prove that

.

Page: 246
Section: 7.C
Number: 3

Suppose

is a positive operator on

and

is a subspace of

invariant under

. Prove that

is a positive operator on

.

Page: 246
Section: 7.C
Number: 4

Suppose

. Prove that

is a positive operator on

and

is a positive operator on

.

Page: 246
Section: 7.C
Number: 232

CHAPTER

Operators on Inner Product Spaces

Page: 246
Section: 7.C
Number: 5

Prove that the sum of two positive operators on

is positive.

Page: 246
Section: 7.C
Number: 6

Suppose

is positive. Prove that

is positive for every positive integer

.

Page: 246
Section: 7.C
Number: 7

Suppose

is a positive operator on

. Prove that

is invertible if and only if

for every

with

.

Page: 246
Section: 7.C
Number: 8

Suppose

. For

, define

by

Prove that

is an inner product on

if and only if

is an invertible positive operator (with respect to the original inner product

).

Page: 246
Section: 7.C
Number: 9

Prove or disprove: the identity operator on

has infinitely many selfadjoint square roots.

Page: 246
Section: 7.C
Number: 10

Suppose

. Prove that the following are equivalent: (a)

is an isometry; (b)

for all

; ©

is an orthonormal list for every orthonormal list of vectors

in

; (d)

is an orthonormal basis for some orthonormal basis

of

.

Page: 246
Section: 7.C
Number: 11

Suppose

are normal operators on

and both operators have 2, 5,

as eigenvalues. Prove that there exists an isometry

such that

.

Page: 246
Section: 7.C
Number: 12

Give an example of two self-adjoint operators

such that the eigenvalues of both operators are 2, 5,7 but there does not exist an isometry

such that

. Be sure to explain why there is no isometry with the required property.

Page: 246
Section: 7.C
Number: 13

Prove or give a counterexample: if

and there exists an orthonormal basis

of

such that

for each

, then

is an isometry.

Page: 246
Section: 7.C
Number: 14

Let

be the second derivative operator in Exercise

in Section 7.A. Show that

is a positive operator.

Page: 253
Section: 7.D
Number: 1

Fix

with

. Define

by

for every

. Prove that

for every

.

Page: 253
Section: 7.D
Number: 2

Give an example of

such that

is the only eigenvalue of

and the singular values of

are

. SECTION 7.D Polar Decomposition and Singular Value Decomposition

Page: 253
Section: 7.D
Number: 3

Suppose

. Prove that there exists an isometry

such that

Page: 253
Section: 7.D
Number: 4

Suppose

and

is a singular value of

. Prove that there exists a vector

such that

and

.

Page: 253
Section: 7.D
Number: 5

Suppose

is defined by

. Find the singular values of

.

Page: 253
Section: 7.D
Number: 6

Find the singular values of the differentiation operator

defined by

, where the inner product on

is as in Example

.

Define

by

Find (explicitly) an isometry

such that

.

Page: 253
Section: 7.D
Number: 8

Suppose

is an isometry, and

is a positive operator such that

. Prove that

. [The exercise above shows that if we write

as the product of an isometry and a positive operator (as in the Polar Decomposition 7.45), then the positive operator equals

.]

Page: 253
Section: 7.D
Number: 9

Suppose

. Prove that

is invertible if and only if there exists a unique isometry

such that

.

Page: 253
Section: 7.D
Number: 10

Suppose

is self-adjoint. Prove that the singular values of

equal the absolute values of the eigenvalues of

, repeated appropriately.

Page: 253
Section: 7.D
Number: 11

Suppose

. Prove that

and

have the same singular values.

Page: 253
Section: 7.D
Number: 12

Prove or give a counterexample: if

, then the singular values of

equal the squares of the singular values of

.

Page: 253
Section: 7.D
Number: 13

Suppose

. Prove that

is invertible if and only if

is not a singular value of

.

Page: 253
Section: 7.D
Number: 14

Suppose

. Prove that dim range

equals the number of nonzero singular values of

.

Page: 253
Section: 7.D
Number: 15

Suppose

. Prove that

is an isometry if and only if all the singular values of

equal

.

Page: 253
Section: 7.D
Number: 240

CHAPTER

Operators on Inner Product Spaces

Page: 253
Section: 7.D
Number: 16

Suppose

. Prove that

and

have the same singular values if and only if there exist isometries

such that

.

Page: 253
Section: 7.D
Number: 17

Suppose

has singular value decomposition given by

for every

, where

are the singular values of

and

are orthonormal bases of

. (a) Prove that if

, then

(b) Prove that if

, then

(d) Suppose

is invertible. Prove that if

, then

for every

.

Page: 253
Section: 7.D
Number: 18

Suppose

. Let

denote the smallest singular value of

, and let

denote the largest singular value of

. (a) Prove that

for every

. (b) Suppose

is an eigenvalue of

. Prove that

.

Page: 253
Section: 7.D
Number: 19

Suppose

. Show that

is uniformly continuous with respect to the metric

on

defined by

.

Page: 253
Section: 7.D
Number: 20

Suppose

. Let

denote the largest singular value of

, let

denote the largest singular value of

, and let

denote the largest singular value of

. Prove that

.

Page: 264
Section: 8.A
Number: 1

Define

by

Find all generalized eigenvectors of

.

Page: 264
Section: 8.A
Number: 2

Define

by

Find the generalized eigenspaces corresponding to the distinct eigenvalues of

.

Page: 264
Section: 8.A
Number: 250

CHAPTER

Operators on Complex Vector Spaces

Page: 264
Section: 8.A
Number: 3

Suppose

is invertible. Prove that

for every

with

.

Page: 264
Section: 8.A
Number: 4

Suppose

and

with

. Prove that

Page: 264
Section: 8.A
Number: 5

Suppose

is a positive integer, and

is such that

but

. Prove that

is linearly independent.

Page: 264
Section: 8.A
Number: 6

Suppose

is defined by

. Prove that

has no square root. More precisely, prove that there does not exist

such that

.

Page: 264
Section: 8.A
Number: 7

Suppose

is nilpotent. Prove that

is the only eigenvalue of

.

Page: 264
Section: 8.A
Number: 8

Prove or give a counterexample: The set of nilpotent operators on

is a subspace of

.

Page: 264
Section: 8.A
Number: 9

Suppose

and

is nilpotent. Prove that

is nilpotent.

Page: 264
Section: 8.A
Number: 10

Suppose that

is not nilpotent. Let

. Show that

.

Page: 264
Section: 8.A
Number: 11

Prove or give a counterexample: If

is a complex vector space and

and

, then

is diagonalizable.

Page: 264
Section: 8.A
Number: 12

Suppose

and there exists a basis of

with respect to which

has an upper-triangular matrix with only

's on the diagonal. Prove that

is nilpotent.

Page: 264
Section: 8.A
Number: 13

Suppose

is an inner product space and

is normal and nilpotent. Prove that

.

Page: 264
Section: 8.A
Number: 14

Suppose

is an inner product space and

is nilpotent. Prove that there exists an orthonormal basis of

with respect to which

has an upper-triangular matrix. [If

, then the result above follows from Schur’s Theorem (6.38) without the hypothesis that

is nilpotent. Thus the exercise above needs to be proved only when

.] SECTION 8.A Generalized Eigenvectors and Nilpotent Operators

Page: 264
Section: 8.A
Number: 15

Suppose

is such that null

. Prove that

is nilpotent and that

for every integer

with

.

Page: 264
Section: 8.A
Number: 16

Suppose

. Show that

Page: 264
Section: 8.A
Number: 17

Suppose

and

is a nonnegative integer such that

Prove that range

for all

.

Page: 264
Section: 8.A
Number: 18

Suppose

. Let

. Prove that

Page: 264
Section: 8.A
Number: 19

Suppose

and

is a nonnegative integer. Prove that null

if and only if

.

Page: 264
Section: 8.A
Number: 20

Suppose

is such that range

. Prove that

is nilpotent.

Page: 264
Section: 8.A
Number: 21

Find a vector space

and

such that null

and range

range

for every positive integer

.

Page: 274
Section: 8.B
Number: 1

Suppose

is a complex vector space,

, and

is the only eigenvalue of

. Prove that

is nilpotent.

Page: 274
Section: 8.B
Number: 2

Give an example of an operator

on a finite-dimensional real vector space such that

is the only eigenvalue of

but

is not nilpotent.

Page: 274
Section: 8.B
Number: 260

CHAPTER

Operators on Complex Vector Spaces

Page: 274
Section: 8.B
Number: 3

Suppose

. Suppose

is invertible. Prove that

and

have the same eigenvalues with the same multiplicities.

Page: 274
Section: 8.B
Number: 4

Suppose

is an

-dimensional complex vector space and

is an operator on

such that null

. Prove that

has at most two distinct eigenvalues.

Page: 274
Section: 8.B
Number: 5

Suppose

is a complex vector space and

. Prove that

has a basis consisting of eigenvectors of

if and only if every generalized eigenvector of

is an eigenvector of

. [For

, the exercise above adds an equivalence to the list in 5.41.]

Page: 274
Section: 8.B
Number: 6

Define

by

Find a square root of

.

Suppose

is a complex vector space. Prove that every invertible operator on

has a cube root.

Page: 274
Section: 8.B
Number: 8

Suppose

and

are eigenvalues of

. Let

. Prove that

null

range

.

Page: 274
Section: 8.B
Number: 9

Suppose

and

are block diagonal matrices of the form

where

has the same size as

for

. Show that

is a block diagonal matrix of the form

Page: 274
Section: 8.B
Number: 10

Suppose

and

. Prove that there exist

such that

, the operator

is diagonalizable,

is nilpotent, and

.

Page: 274
Section: 8.B
Number: 11

Suppose

and

. Prove that for every basis of

with respect to which

has an upper-triangular matrix, the number of times that

appears on the diagonal of the matrix of

equals the multiplicity of

as an eigenvalue of

. SECTION 8.C Characteristic and Minimal Polynomials

Page: 274
Section: 8.B
Number: 261

8.C Characteristic and Minimal Polynomials The Cayley-Hamilton Theorem The next definition associates a polynomial with each operator on

if

. For

, the corresponding definition will be given in the next chapter. 8.34 Definition characteristic polynomial Suppose

is a complex vector space and

. Let

denote the distinct eigenvalues of

, with multiplicities

. The polynomial

is called the characteristic polynomial of

. 8.35 Example Suppose

is defined as in Example 8.25. Because the eigenvalues of

are

, with multiplicity

, and

, with multiplicity

, we see that the characteristic polynomial of

is

. 8.36 Degree and zeros of characteristic polynomial Suppose

is a complex vector space and

. Then (a) the characteristic polynomial of

has degree

; (b) the zeros of the characteristic polynomial of

are the eigenvalues of

. Proof Clearly part (a) follows from

and part (b) follows from the definition of the characteristic polynomial. Most texts define the characteristic polynomial using determinants (the two definitions are equivalent by 10.25). The approach taken here, which is considerably simpler, leads to the following easy proof of the CayleyHamilton Theorem. In the next chapter, we will see that this result also holds on real vector spaces (see 9.24). 8.37 Cayley-Hamilton Theorem Suppose

is a complex vector space and

. Let

denote the characteristic polynomial of

. Then

.

Page: 282
Section: 8.C
Number: 1

Suppose

is such that the eigenvalues of

are 3, 5, 8. Prove that

.

Page: 282
Section: 8.C
Number: 2

Suppose

is a complex vector space. Suppose

is such that

and

are eigenvalues of

and that

has no other eigenvalues. Prove that

, where

.

Page: 282
Section: 8.C
Number: 3

Give an example of an operator on

whose characteristic polynomial equals

.

Page: 282
Section: 8.C
Number: 268

CHAPTER

Operators on Complex Vector Spaces

Page: 282
Section: 8.C
Number: 4

Give an example of an operator on

whose characteristic polynomial equals

and whose minimal polynomial equals

.

Page: 282
Section: 8.C
Number: 5

Give an example of an operator on

whose characteristic and minimal polynomials both equal

.

Page: 282
Section: 8.C
Number: 6

Give an example of an operator on

whose characteristic polynomial equals

and whose minimal polynomial equals

.

Page: 282
Section: 8.C
Number: 7

Suppose

is a complex vector space. Suppose

is such that

. Prove that the characteristic polynomial of

is

, where

null

and

.

Page: 282
Section: 8.C
Number: 8

Suppose

. Prove that

is invertible if and only if the constant term in the minimal polynomial of

is nonzero.

Page: 282
Section: 8.C
Number: 9

Suppose

has minimal polynomial

. Find the minimal polynomial of

.

Page: 282
Section: 8.C
Number: 10

Suppose

is a complex vector space and

is invertible. Let

denote the characteristic polynomial of

and let

denote the characteristic polynomial of

. Prove that

for all nonzero

.

Page: 282
Section: 8.C
Number: 11

Suppose

is invertible. Prove that there exists a polynomial

such that

.

Page: 282
Section: 8.C
Number: 12

Suppose

is a complex vector space and

. Prove that

has a basis consisting of eigenvectors of

if and only if the minimal polynomial of

has no repeated zeros. [For complex vector spaces, the exercise above adds another equivalence to the list given by 5.41.]

Page: 282
Section: 8.C
Number: 13

Suppose

is an inner product space and

is normal. Prove that the minimal polynomial of

has no repeated zeros.

Page: 282
Section: 8.C
Number: 14

Suppose

is a complex inner product space and

is an isometry. Prove that the constant term in the characteristic polynomial of

has absolute value

. SECTION 8.C Characteristic and Minimal Polynomials

Page: 282
Section: 8.C
Number: 15

Suppose

and

. (a) Prove that there exists a unique monic polynomial

of smallest degree such that

. (b) Prove that

divides the minimal polynomial of

.

Page: 282
Section: 8.C
Number: 16

Suppose

is an inner product space and

. Suppose

is the minimal polynomial of

. Prove that

is the minimal polynomial of

.

Page: 282
Section: 8.C
Number: 17

Suppose

and

. Suppose the minimal polynomial of

has degree

. Prove that the characteristic polynomial of

equals the minimal polynomial of

.

Page: 282
Section: 8.C
Number: 18

Suppose

. Find the minimal and characteristic polynomials of the operator on

whose matrix (with respect to the standard basis) is

[The exercise above shows that every monic polynomial is the characteristic polynomial of some operator.]

Page: 282
Section: 8.C
Number: 19

Suppose

is a complex vector space and

. Suppose that with respect to some basis of

the matrix of

is upper triangular, with

on the diagonal of this matrix. Prove that the characteristic polynomial of

is

.

Page: 282
Section: 8.C
Number: 20

Suppose

is a complex vector space and

are nonzero subspaces of

such that

. Suppose

and each

is invariant under

. For each

, let

denote the characteristic polynomial of

. Prove that the characteristic polynomial of

equals

Page: 289
Section: 8.D
Number: 1

Find the characteristic polynomial and the minimal polynomial of the operator

in Example 8.53.

Page: 289
Section: 8.D
Number: 2

Find the characteristic polynomial and the minimal polynomial of the operator

in Example 8.54.

Page: 289
Section: 8.D
Number: 3

Suppose

is nilpotent. Prove that the minimal polynomial of

is

, where

is the length of the longest consecutive string of

's that appears on the line directly above the diagonal in the matrix of

with respect to any Jordan basis for

.

Page: 289
Section: 8.D
Number: 4

Suppose

and

is a basis of

that is a Jordan basis for

. Describe the matrix of

with respect to the basis

obtained by reversing the order of the

's.

Page: 289
Section: 8.D
Number: 5

Suppose

and

is a basis of

that is a Jordan basis for

. Describe the matrix of

with respect to this basis.

Page: 289
Section: 8.D
Number: 6

Suppose

is nilpotent and

and

are as in 8.55. Prove that

is a basis of null

. [The exercise above implies that

, which equals dim null

, depends only on

and not on the specific Jordan basis chosen for

.]

Suppose

are monic polynomials with the same zeros and

is a polynomial multiple of

. Prove that there exists

such that the characteristic polynomial of

is

and the minimal polynomial of

is

.

Page: 289
Section: 8.D
Number: 8

Suppose

is a complex vector space and

. Prove that there does not exist a direct sum decomposition of

into two proper subspaces invariant under

if and only if the minimal polynomial of

is of the form

for some

.

Page: 300
Section: 9.A
Number: 1

Prove 9.3.

Page: 300
Section: 9.A
Number: 2

Verify that if

is a real vector space and

, then

.

Page: 300
Section: 9.A
Number: 3

Suppose

is a real vector space and

. Prove that

is linearly independent in

if and only if

is linearly independent in

.

Page: 300
Section: 9.A
Number: 4

Suppose

is a real vector space and

. Prove that

spans

if and only if

spans

.

Page: 300
Section: 9.A
Number: 5

Suppose that

is a real vector space and

. Show that

and that

for every

.

Page: 300
Section: 9.A
Number: 6

Suppose

is a real vector space and

. Prove that

is invertible if and only if

is invertible.

Page: 300
Section: 9.A
Number: 7

Suppose

is a real vector space and

. Prove that

is nilpotent if and only if

is nilpotent.

Page: 300
Section: 9.A
Number: 8

Suppose

and

are eigenvalues of

. Prove that

has no nonreal eigenvalues.

Page: 300
Section: 9.A
Number: 9

Prove there does not exist an operator

such that

is nilpotent.

Page: 300
Section: 9.A
Number: 10

Give an example of an operator

such that

is nilpotent.

Page: 300
Section: 9.A
Number: 286

CHAPTER

Operators on Real Vector Spaces

Page: 300
Section: 9.A
Number: 11

Suppose

is a real vector space and

. Suppose there exist

such that

. Prove that

has an eigenvalue if and only if

.

Page: 300
Section: 9.A
Number: 12

Suppose

is a real vector space and

. Suppose there exist

such that

and

is nilpotent. Prove that

has no eigenvalues.

Page: 300
Section: 9.A
Number: 13

Suppose

is a real vector space,

, and

are such that

. Prove that null

has even dimension for every positive integer

.

Page: 300
Section: 9.A
Number: 14

Suppose

is a real vector space with

. Suppose

is such that

is nilpotent. Prove that

.

Page: 300
Section: 9.A
Number: 15

Suppose

is a real vector space and

has no eigenvalues. Prove that every subspace of

invariant under

has even dimension.

Page: 300
Section: 9.A
Number: 16

Suppose

is a real vector space. Prove that there exists

such that

if and only if

has even dimension.

Page: 300
Section: 9.A
Number: 17

Suppose

is a real vector space and

satisfies

. Define complex scalar multiplication on

as follows: if

, then

(a) Show that the complex scalar multiplication on

defined above and the addition on

makes

into a complex vector space. (b) Show that the dimension of

as a complex vector space is half the dimension of

as a real vector space.

Page: 300
Section: 9.A
Number: 18

Suppose

is a real vector space and

. Prove that the following are equivalent: (a) All the eigenvalues of

are real. (b) There exists a basis of

with respect to which

consisting of generalized eigenvectors of

.

Page: 300
Section: 9.A
Number: 19

Suppose

is a real vector space with

and

is such that null

. Prove that

has at most two distinct eigenvalues and that

has no nonreal eigenvalues.

Page: 309
Section: 9.B
Number: 1

Suppose

is an isometry. Prove that there exists a nonzero vector

such that

.

Page: 309
Section: 9.B
Number: 2

Prove that every isometry on an odd-dimensional real inner product space has

or

as an eigenvalue.

Page: 309
Section: 9.B
Number: 3

Suppose

is a real inner product space. Show that

for

defines a complex inner product on

.

Page: 309
Section: 9.B
Number: 4

Suppose

is a real inner product space and

is self-adjoint. Show that

is a self-adjoint operator on the inner product space

defined by the previous exercise.

Page: 309
Section: 9.B
Number: 5

Use the previous exercise to give a proof of the Real Spectral Theorem (7.29) via complexification and the Complex Spectral Theorem (7.24).

Page: 309
Section: 9.B
Number: 6

Give an example of an operator

on an inner product space such that

has an invariant subspace whose orthogonal complement is not invariant under

. [The exercise above shows that

can fail without the hypothesis that

is normal.]

Suppose

and

has a block diagonal matrix

with respect to some basis of

. For

, let

be the operator on

whose matrix with respect to the same basis is a block diagonal matrix with blocks the same size as in the matrix above, with

in the

block, and with all the other blocks on the diagonal equal to identity matrices (of the appropriate size). Prove that

.

Page: 309
Section: 9.B
Number: 8

Suppose

is the differentiation operator on the vector space

in Exercise

in Section 7.A. Find an orthonormal basis of

such that the matrix of the normal operator

has the form promised by

.

Page: 319
Section: 10.A
Number: 1

Suppose

and

is a basis of

. Prove that the matrix

is invertible if and only if

is invertible.

Page: 319
Section: 10.A
Number: 2

Suppose

and

are square matrices of the same size and

. Prove that

.

Page: 319
Section: 10.A
Number: 3

Suppose

has the same matrix with respect to every basis of

. Prove that

is a scalar multiple of the identity operator.

Page: 319
Section: 10.A
Number: 4

Suppose

and

are bases of

. Let

be the operator such that

for

. Prove that

Page: 319
Section: 10.A
Number: 5

Suppose

is a square matrix with complex entries. Prove that there exists an invertible square matrix

with complex entries such that

is an upper-triangular matrix.

Page: 319
Section: 10.A
Number: 6

Give an example of a real vector space

and

such that

.

Page: 319
Section: 10.A
Number: 7

Suppose

is a real vector space,

, and

has a basis consisting of eigenvectors of

. Prove that

. SECTION 10.A Trace

Page: 319
Section: 10.A
Number: 8

Suppose

is an inner product space and

. Define

by

. Find a formula for trace

.

Page: 319
Section: 10.A
Number: 9

Suppose

satisfies

. Prove that

Page: 319
Section: 10.A
Number: 10

Suppose

is an inner product space and

. Prove that

Page: 319
Section: 10.A
Number: 11

Suppose

is an inner product space. Suppose

is a positive operator and trace

. Prove that

.

Page: 319
Section: 10.A
Number: 12

Suppose

is an inner product space and

are orthogonal projections. Prove that

.

Page: 319
Section: 10.A
Number: 13

Suppose

is the operator whose matrix is

Someone tells you (accurately) that

and

are eigenvalues of

. Without using a computer or writing anything down, find the third eigenvalue of

.

Page: 319
Section: 10.A
Number: 14

Suppose

and

. Prove that

trace

.

Page: 319
Section: 10.A
Number: 15

Suppose

. Prove that

.

Page: 319
Section: 10.A
Number: 16

Prove or give a counterexample: if

, then

trace

.

Page: 319
Section: 10.A
Number: 17

Suppose

is such that

for all

. Prove that

.

Page: 319
Section: 10.A
Number: 18

Suppose

is an inner product space with orthonormal basis

and

. Prove that

Conclude that the right side of the equation above is independent of which orthonormal basis

is chosen for

.

Page: 319
Section: 10.A
Number: 306

CHAPTER

Trace and Determinant

Page: 319
Section: 10.A
Number: 19

Suppose

is an inner product space. Prove that

defines an inner product on

.

Page: 319
Section: 10.A
Number: 20

Suppose

is a complex inner product space and

. Let

be the eigenvalues of

, repeated according to multiplicity. Suppose

is the matrix of

with respect to some orthonormal basis of

. Prove that

Page: 319
Section: 10.A
Number: 21

Suppose

is an inner product space. Suppose

and

for every

. Prove that

is normal. [The exercise above fails on infinite-dimensional inner product spaces, leading to what are called hyponormal operators, which have a welldeveloped theory.]

Page: 345
Section: 10.B
Number: 1

Suppose

is a real vector space. Suppose

has no eigenvalues. Prove that

.

Page: 345
Section: 10.B
Number: 2

Suppose

is a real vector space with even dimension and

. Suppose det

. Prove that

has at least two distinct eigenvalues.

Page: 345
Section: 10.B
Number: 3

Suppose

and

. Let

denote the eigenvalues of

(or of

if

is a real vector space), repeated according to multiplicity. (a) Find a formula for the coefficient of

in the characteristic polynomial of

in terms of

. (b) Find a formula for the coefficient of

in the characteristic polynomial of

in terms of

. SECTION 10.B Determinant

Page: 345
Section: 10.B
Number: 4

Suppose

and

. Prove that

.

Page: 345
Section: 10.B
Number: 5

Prove or give a counterexample: if

, then

.

Page: 345
Section: 10.B
Number: 6

Suppose

is a block upper-triangular matrix

where each

along the diagonal is a square matrix. Prove that

Page: 345
Section: 10.B
Number: 7

Suppose

is an

-by-

matrix with real entries. Let

denote the operator on

whose matrix equals

, and let

denote the operator on

whose matrix equals

. Prove that trace

and

.

Page: 345
Section: 10.B
Number: 8

Suppose

is an inner product space and

. Prove that

Use this to prove that

, giving a different proof than was given in

.

Page: 345
Section: 10.B
Number: 9

Suppose

is an open subset of

and

is a function from

to

. Suppose

and

is differentiable at

. Prove that the operator

satisfying the equation in

is unique. [This exercise shows that the notation

is justified.]

Page: 345
Section: 10.B
Number: 10

Suppose

and

. Prove that

is differentiable at

and

.

Page: 345
Section: 10.B
Number: 11

Find a suitable hypothesis on

and then prove

.

Page: 345
Section: 10.B
Number: 12

Let

be positive numbers. Find the volume of the ellipsoid

by finding a set

whose volume you know and an operator

such that

equals the ellipsoid above.